In selecting one boy's share, take the 2 parts from one apple, or 1 part from each of the two apples; hence,

expresses either 2 thirds of 1 thing, or 1 third of 2 things. Also, expresses 3 fifths of 1 thing, or 1 fifth of 3 things. Therefore, the numerator of a fraction expresses the number of units to be divided; and the denominator the divisor, or what part is taken from each. Hence,

A fraction is expressed in the form of an unexecuted division,
In which, The DIVIDEND is the NUMERATOR;
The DIVISOR is the DENOMINATOR;
The QUOTIENT is the FRACTION itself.

is the quotient of 1 (numerator)÷3 (denum.);
is the quotient of 4 (numerator) ÷ 5 (denom.);
is the quotient of 7 (numerator)÷6 (denom.).

ART. 126. Since fractions arise from division, one of the signs of division (Art. 20,) is used in expressing them; the numerator being written above, and the denominator below, a horizontal line.

Thus, three-fifths is written ; two-sixths is written 2.

TO READ COMMON FRACTIONS.

Read the number of parts taken, as expressed by the numerator; then the size of the parts, as expressed by the denominator.

TO WRITE COMMON FRACTIONS.

Write the numerator, place a horizontal line below it, under which write the denominator.

REM. In reading, means two-thirds of one.

There are two other methods, (Art. 125): thus, may be read, one-third of two, or two divided by three; but these methods are rarely used.

FRACTIONS TO BE READ.

6 14
45

8

9, §, 1, 10, 14, 13, 5, 2, 13, 18, 10, 180, 240, 124

1243°

REVIEW.-125. What do two-thirds express? What does the numerator of a fraction express? The denominator? In what form is a fraction expressed? What is the dividend? The divisor? The quotient? Give examples. 126. What sign is used in fractions? Why? How is a common fraction read? Give examples. How written? Give examples.

Twenty-three-one-hundred-and-fourths.

Seven-elevenths: Eleven-twentieths:

If an orange be cut into 8 equal parts, what fraction will express 3 of the parts?

If 3 apples be divided equally among 4 boys, what part of an apple will be given to each boy?

What part of 1 apple, is a third part of 2 apples?
What expresses the quotient of 5, divided by 8?

ART. 127. A whole number may be expressed in the form of a fraction, by writing 1 below it for a denominator. Thus, 2 may be written and is read two-ones. 3 may be written and is read three-ones. and is read four-ones.

4 may be written

But, 2 ones are 2; 3 ones, 3, &c.; hence, the value of the number is not changed.

ART. 128. If 2 apples be divided, each into four equal parts, there will be 8 parts in all. Three of the parts (fourths) are expressed by ; 4 parts by 4; 5 parts by 5.

When the number of parts taken is less than the number into which the unit is divided, the value expressed is less than one, or the whole thing;

When the number of parts taken is equal to the number into which the unit is divided, the value, as, is equal to 1; When greater than the number into which the unit is divided, the value,, is greater than 1. Hence,

REVIEW.-126. Of the fractions to be read, which expresses parts of the largest size? Which the smallest? Which the least number? Which the greatest? Which the same? Which parts of the same size?

127. How may a whole number be expressed in fractional form? Does this change its value? Why not? 128. When is the value of a fraction less than 1? When equal? When greater? Illustrate by examples,

1st. When the numerator is less than the denominator, the value of the fraction is less than 1.

2d. When the numerator is equal to the denominator, the value of the fraction is equal to 1.

3d. When the numerator is greater than the denominator, the value of the fraction is greater than 1.

DEFINITIONS.

ART. 129. 1. A Fraction is an expression of one or more of the equal parts of a unit, or one thing.

2. A Proper Fraction has a numerator less than the denominator; as, 1, 3, and §.

3. An Improper Fraction has a numerator equal to, or greater than the denominator; as, and 5.

3)

REM.-A proper fraction is so termed, because it expresses a value less than one. An improper fraction is not properly a fraction of a unit, the value expressed being equal to, or greater than one.

4. A Simple Fraction is a single fraction, either proper or improper; as,,, and .

5. A Compound Fraction is a fraction of a fraction, or several fractions joined by of; as, of, of 3 of 4.

6. A Mixed Number is a fraction joined to a whole number; as, 21, 31, and 57.

7. A Complex Fraction has a fraction in one or in both of

ART. 130. If a line,

as A B, be divided into

two equal parts, one of the parts, as A C, is termed onehalf (): that is, one-half of 1, or the whole thing.

REVIEW.-129. What is a fraction? A proper fraction? An improper fraction? REM. Why is a proper fraction so termed? An improper fraction? What is a simple fraction? A compound fraction? A mixed number? A complex fraction? Give examples of each. 130. What is the half of one-half? Why? The third of one-half? Why?

If a half be divided into 3 equal parts, as in the figure, one of the parts is one-third of one-half, expressed thus, of; and this is one-sixth of the whole line: that is, of is

In the same manner, it may be shown, that of is 1; that theofis; that of is; that ofis ; &c. 142

If one apple be divided into three equal parts, and each of these parts into 3 other equal parts, into how many parts will the whole be divided?

18

What part of the whole will one piece be? What is of }?

If one orange be divided into 3 equal parts, and each part into 4 other equal parts, into how many equal parts will the whole be divided?

183

What part of the whole will 1 piece be? What is of ? What is of; of? of? of? of? Young Pupils may omit the Illustrations until they review the book.

GENERAL PRINCIPLES.

ART. 131. If the numerator of the fraction be multiplied by 3, without changing the denominator, the result will be .

ILLUSTRATION.-Each of the fractions and having the same denominator, expresses parts of the same size: but, as the numerator of the second fraction (), is 3 times that of the first, (), it expresses 3 times as many of those equal parts as the first, and is 3 times as large. The same may be shown of any other fraction. Hence,

PROPOSITION 1.-If the numerator be multiplied, without changing the denominator, the value of the fraction will be multiplied as many times as there are units in the multiplier.

Hence, a fraction is multiplied, by multiplying its numerator.

REVIEW.-131. What is the effect of multiplying the numerator of a fraction, without changing the denominator, Prop. I?

ART. 132. If the numerator of the fraction be divided by 2, without changing the denominator, the

result will be .

denominator, expresses parts of the same size; but the numerator of the second fraction (3), is only one-half the numerator of the first (); therefore, expresses one-half as many of those equal parts as the first (4), and is one-half the value. Hence,

PROP. II.-If the numerator be divided without changing the denominator, the value of the fraction will be divided as many times as there are units in the divisor.

Hence, a fraction is divided by dividing its numerator.

ART. 133. If the denominator of the fraction be multiplied by 2, without changing the numerator, the result will be .

same numerator, denotes the same number of parts; but, in the second (3) the parts are one-half the size of those in the first (3): but, of (Art. 130); consequently, the whole value of the second fraction is one-half that of the first. Hence,

PROP. III. If the denominator be multiplied without changing the numerator, the value of the fraction will be divided as many times as there are units in the multiplier.

Hence, a fraction is divided, by multiplying its denominator.

ART. 134. If the denominator of the fraction be divided by 3, without changing the numerator, the result will be.

ILLUSTRATION. Each of the fractions and having the same numerator, denotes the same number of parts; but, in the